Theorem onexlimgt 43688

Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexlimgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9558	. . . 4 ⊢ ω ∈ On
2		onun2 6420	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
3	1, 2	mpan2 697	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4		onexomgt 43686	. . 3 ⊢ ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6		simp2 1143	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7		omcl 8461	. . . . 5 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
8	1, 6, 7	sylancr 593	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9		noel 4266	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∪ ω) ∈ ∅
10		oveq2 7364	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11		om0 8442	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → (ω ·o ∅) = ∅)
12	1, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ω ·o ∅) = ∅
13	10, 12	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
14	13	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
15	14	notbid 319	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
16	15	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
17	9, 16	mpbiri 259	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
18	17	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
19	18	ex 413	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
20	19	com23 86	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21	20	3impia 1123	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22		limom 7822	. . . . . . . . 9 ⊢ Lim ω
23	1, 22	pm3.2i 471	. . . . . . . 8 ⊢ (ω ∈ On ∧ Lim ω)
24	6, 23	jctir 525	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25		omlimcl2 43687	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
26	24, 25	sylan 586	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
27	26	ex 413	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28		on0eqel 6435	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
29	6, 28	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
30	21, 27, 29	mpjaod 866	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31		simp1 1142	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
32	31, 8	jca 516	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33		simp3 1144	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34		ssun1 4107	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ ω)
35	33, 34	jctil 524	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36		ontr2 6358	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
37	32, 35, 36	sylc 65	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38		limeq 6322	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39		eleq2 2828	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (ω ·o 𝑎)))
40	38, 39	anbi12d 638	. . . . 5 ⊢ (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
41	40	rspcev 3560	. . . 4 ⊢ (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
42	8, 30, 37, 41	syl12anc 842	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
43	42	rexlimdv3a 3144	. 2 ⊢ (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥)))
44	5, 43	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  Oncon0 6310  Lim wlim 6311  (class class class)co 7356  ωcom 7806   ·_o comu 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400

This theorem is referenced by: (None)